A000714 Number of partitions of n, with three kinds of 1 and 2 and two kinds of 3,4,5,.... (Formerly M2777 N1117)

1, 3, 9, 21, 47, 95, 186, 344, 620, 1078, 1835, 3045, 4967, 7947, 12534, 19470, 29879, 45285, 67924, 100820, 148301, 216199, 312690, 448738, 639464, 905024, 1272837, 1779237, 2473065, 3418655, 4701611, 6434015, 8763676

Offset

0,2

Comments

Convolution of A000712 and A008619. - Vaclav Kotesovec, Aug 18 2015

References

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

T. Doslic, Kepler-Bouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.

N. J. A. Sloane, Transforms

Formula

EULER transform of 3, 3, 2, 2, 2, 2, 2, 2, ...

G.f.: 1/((1-x)*(1-x^2)*Product_{k>=1} (1 - x^k)^2). - Emeric Deutsch, Apr 17 2006

a(n) ~ 3^(1/4) * exp(2*Pi*sqrt(n/3)) / (8 * Pi^2 * n^(1/4)). - Vaclav Kotesovec, Aug 18 2015

Example

a(2)=9 because we have 2, 2', 2", 1+1, 1'+1', 1"+1", 1+1', 1+1", 1'+1".

Maple

g:=1/((1-x)*(1-x^2)*product((1-x^k)^2, k=1..40)): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=0..32); # Emeric Deutsch, Apr 17 2006

Mathematica

p=Product[1/(1-x^i), {i, 1, 20}]; CoefficientList[Series[p^2/(1 - x)/(1 - x^2), {x, 0, 20}], x] (* Geoffrey Critzer, Nov 28 2011 *)

Crossrefs

Sequence in context: A141156 A262197 A014286 * A267226 A273845 A090984

Adjacent sequences: A000711 A000712 A000713 * A000715 A000716 A000717

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Extended with formula from Christian G. Bower, Apr 15 1998

Status

approved